2021 Fall AMC 12B Problems/Problem 12
Problem 12
Let 
What is the value of
![\[\frac{\sin 3c \cdot \sin 6c \cdot \sin 9c \cdot \sin 12c \cdot \sin 15c}{\sin c \cdot \sin 2c \cdot \sin 3c \cdot \sin 4c \cdot \sin 5c}?\]](http://latex.artofproblemsolving.com/5/c/b/5cb66360bc69b72e86eac3564c26bae7fd577bc9.png)
Solution 1
The prime factorization of 
is 
and the prime factorization of 
is 
so
![\[f(768)=(1+\frac{1}{2}+\ldots+\frac{1}{256})(1+\frac{1}{3})=\frac{511}{192}\]](http://latex.artofproblemsolving.com/e/5/c/e5c18015481892cb263aa4715af8908e1fd60b87.png)
![\[f(384)=(1+\frac{1}{2}+\ldots+\frac{1}{128})(1+\frac{1}{3})=\frac{510}{192}\]](http://latex.artofproblemsolving.com/6/4/a/64acddaee95d9825e1506ba4cf483f39f2d526c8.png)
so the difference is 
~lopkiloinm
See Also
| 2021 Fall AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 11 |
Followed by Problem 13 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
